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The Nematic Phase of a System of Long Hard Rods

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Abstract

We consider a two-dimensional lattice model for liquid crystals consisting of long rods interacting via purely hard core interactions, with two allowed orientations defined by the underlying lattice. We rigorously prove the existence of a nematic phase, i.e., we show that at intermediate densities the system exhibits orientational order, either horizontal or vertical, but no positional order. The proof is based on a two-scales cluster expansion: we first coarse grain the system on a scale comparable with the rods’ length; then we express the resulting effective theory as a contour’s model, which can be treated by Pirogov-Sinai methods.

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References

  1. Angelescu N., Zagrebnov V.A.: A Lattice Model of Liquid Crystals with Matrix Order Parameter. J. Phys. A 15, L639–L642 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  2. Angelescu N., Romano S., Zagrebnov V.A.: On Long-Range Order in Low-Dimensional Lattice-Gas Models of Nematic Liquid Crystals. Phys. Lett. A 200, 433–437 (1995)

    Article  ADS  Google Scholar 

  3. Blinc, R., Zeks, B.: Soft Modes in Ferroelectrics and Antiferroelectrics. Amsterdam: North-Holland, (1974)

  4. Borgs C., Imbrie J.Z.: A Unified Approach to Phase Diagrams in Field Theory and Statistical Mechanics. Commun. Math. Phys. 123, 305–328 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Bricmont J., Kuroda K., Lebowitz J.L.: The structure of Gibbs states and phase coexistence for nonsymmetric continuum Widom-Rowlinson models. Z. Wahrsch. Verw. Geb. 67, 121–138 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brydges, D.C.: A short course on cluster expansions. In: K. Osterwalder, R. Stora, eds., “Critical Phenomena, Random Systems, Gauge Theories”, Les Houches Summer School, Amsterdam-New York:North Holland, 1984, pp. 131183

  7. de Gennes P.G., Prost J.: The Physics of Liquid Crystals. Oxford University Press, Oxford (1993)

    Google Scholar 

  8. Dhar D., Rajesh R., Stilck J.F.: Hard rigid rods on a Bethe-like lattice. Phys. Rev. E 84, 011140 (2011)

    Article  ADS  Google Scholar 

  9. Dyson F.J., Lieb E.H., Simon B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–383 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  10. Fischer T., Vink R.L.C.: Restricted orientation “liquid crystal” in two dimensions: Isotropic-nematic transition or liquid-gas one (?). Europhys. Lett. 85, 56003 (2009)

    Article  ADS  Google Scholar 

  11. Fröhlich J., Israel R., Lieb E.H., Simon B.: Phase transitions and reflection positivity. I. General theory and long range lattice models. Commun. Math. Phys. 62, 1–34 (1978)

    Article  ADS  Google Scholar 

  12. Fröhlich J., Simon B., Spencer T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50, 79–95 (1976)

    Article  ADS  Google Scholar 

  13. Fröhlich J., Spencer T.: The KosterlitzThouless transition in two-dimensional abelian systems and the Coulomb gas. Commun. Math. Phys. 81, 527602 (1981)

    Google Scholar 

  14. Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of ergodic, qualitative, and statistical theory of motion. Berlin-Heidelberg-New York: Springer, 2004

  15. Ghosh A., Dhar D.: On the orientational ordering of long rods on a lattice. Europhys. Lett. 78, 20003 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  16. Gruber C., Griffiths R.B.: Phase transition in a ferromagnetic fluid. Physica A 138, 220230 (1986)

    Article  MathSciNet  Google Scholar 

  17. Gruber C., Tamura H., Zagrebnov V.A.: Berezinskii–Kosterlitz–Thouless Order in Two-Dimensional O(2)-Ferrofluid. J. Stat. Phys. 106, 875–893 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Heilmann O.J.: Existence of phase transition in certain lattice gases with repulsive potential. Lett. Nuovo Cim. 3, 95 (1972)

    Article  Google Scholar 

  19. Heilmann, O.J., Lieb, E.H.: Monomers and Dimers. Phys. Rev. Lett. 24, 1412 (1970); Theory of Monomer-Dimer systems. Commun. Math. Phys. 25, 190–232 (1972)

    Google Scholar 

  20. Heilmann O.J., Lieb E.H.: Lattice Models for Liquid Crystals. J. Stat. Phys. 20, 679–693 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  21. Huckaby D.A.: Phase transitions in lattice gases of hard-core molecules having two orientations. J. Stat. Phys. 17, 371–375 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  22. Ioffe D., Velenik Y., Zahradnik M.: Entropy-Driven Phase Transition in a Polydisperse Hard-Rods Lattice System. J. Stat. Phys. 122, 761–786 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Kotecky, R.: Pirogov-Sinai Theory. In: J.-P. Francoise, G.L. Naber, T.S. Tsun, eds, Encyclopedia of Mathematical Physics, Oxford: Elsiever, 2006, pp. 60–65

  24. Kundu, J., Rajesh, R., Dhar, D., Stilck, J.F.: The nematic-disordered phase transition in systems of long rigid rods on two dimensional lattices. Phys. Rev. E 87, 032103 (2013)

    Google Scholar 

  25. Lebowitz J.L., Gallavotti G.: Phase transitions in binary lattice gases. J. Math. Phys. 12, 1129–1133 (1971)

    Article  ADS  Google Scholar 

  26. Lebowitz J.L., Penrose O.: Rigorous Treatment of the Van Der Waals Van Der Walls Maxwell Theory of the Liquid-Vapor Transition. J. Math. Phys. 7, 98–113 (1966)

    Article  MathSciNet  ADS  Google Scholar 

  27. Letawe, I.: Le module de cristaux liquides de Heilmann et Lieb. Mémoire de Licenciée en Sciences, Louvain-la-Neuve:Université Catholique de Louvain, 1994

  28. Lopez L.G., Linares D.H., Ramirez-Pastor A.J.: Critical exponents and universality for the isotropic-nematic phase transition in a system of self-assembled rigid rods on a lattice. Phys. Rev. E 80, 040105(R) (2009)

    Article  ADS  Google Scholar 

  29. Lopez L.G., Linares D.H., Ramirez-Pastor A.J., Cannas S.A.: Phase diagram of self-assembled rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations. J. Chem. Phys. 133, 134706 (2010)

    Article  ADS  Google Scholar 

  30. Matoz-Fernandez D.A., Linares D.H., Ramirez-Pastor A.J.: Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations. J. Chem. Phys. 128, 214902 (2008)

    Article  ADS  Google Scholar 

  31. Matoz-Fernandez D.A., Linares D.H., Ramirez-Pastor A.J.: Determination of the critical exponents for the isotropic-nematic phase transition in a system of long rods on two-dimensional lattices: Universality of the transition. Europhys. Lett. 82, 50007 (2008)

    Article  ADS  Google Scholar 

  32. Matoz-Fernandez D.A., Linares D.H., Ramirez-Pastor A.J.: Critical behavior of long linear k-mers on honeycomb lattices. Phys. A 387, 6513–6525 (2008)

    Article  Google Scholar 

  33. Maier W., Saupe A.: A simple molecular statistical theory of the nematic crystalline-liquid phase. Z. Naturf. 14(A), 882–889 (1959)

    ADS  Google Scholar 

  34. Onsager L.: The effects of shape on the interaction of colloidal particles. Ann. N. Y. Acad. Sci. 51, 627–659 (1949)

    Article  ADS  Google Scholar 

  35. Parisi G., Zamponi F.: Mean-field theory of hard sphere glasses and jamming. Rev. Mod. Phys. 82, 789–845 (2010)

    Article  ADS  Google Scholar 

  36. Pikin S.A.: Structural Transitions in Liquid Crystals. Nauka, Moscow (1981)

    Google Scholar 

  37. Pirogov, S., Sinai, Ya.: Phase diagrams of classical lattice systems. Theor. Math. Phys. 25, 1185–1192 (1975) and 26, 39–49 (1976)

    Google Scholar 

  38. Ruelle D.: Existence of a Phase Transition in a Continuous Classical System. Phys. Rev. Lett. 27, 1040–1041 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  39. Ruelle D.: Statistical mechanics: rigorous results. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

  40. Zagrebnov V.A.: Long-range order in a lattice-gas model of nematic liquid crystals. Physica A 232, 737–746 (1996)

    Article  ADS  Google Scholar 

  41. Zahradnik M.: An alternative version of Pirogov-Sinai theory. Commun. Math. Phys. 93, 559–581 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  42. Zahradnik M.: A short course on the Pirogov-Sinai theory. Rendiconti Math. Serie VII 18, 411–486 (1998)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Av. de l’Université, BP.12, Technopôle du Madrillet, 76801, Saint-Étienne-du-Rouvray, France

    Margherita Disertori

  2. Dipartimento di Matematica e Fisica, Università di Roma Tre, L.go S. L. Murialdo 1, 00146, Roma, Italy

    Alessandro Giuliani

Authors
  1. Margherita Disertori
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  2. Alessandro Giuliani
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Correspondence to Alessandro Giuliani.

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Communicated by H. Spohn

Dedicated to the 70 th birthday of Giovanni Gallavotti

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Disertori, M., Giuliani, A. The Nematic Phase of a System of Long Hard Rods. Commun. Math. Phys. 323, 143–175 (2013). https://doi.org/10.1007/s00220-013-1767-1

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